Ursescu theorem

Generalization of closed graph, open mapping, and uniform boundedness theorem

Theorem^[1] (Ursescu) — Let ${\displaystyle X}$ be a complete semi-metrizable locally convex topological vector space and ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ be a closed convex multifunction with non-empty domain. Assume that ${\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}$ is a barrelled space for some/every ${\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}$ Assume that ${\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}$ and let ${\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)}$ (so that ${\displaystyle y_{0}\in {\mathcal {R}}\left(x_{0}\right)}$). Then for every neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ in ${\displaystyle X,}$ ${\displaystyle y_{0}}$ belongs to the relative interior of ${\displaystyle {\mathcal {R}}(U)}$ in ${\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}$ (that is, ${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}$). In particular, if ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }$ then ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}$

Closed graph theorem — Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Fréchet spaces and ${\displaystyle T:X\to Y}$ be a linear map. Then ${\displaystyle T}$ is continuous if and only if the graph of ${\displaystyle T}$ is closed in ${\displaystyle X\times Y.}$

Proof

For the non-trivial direction, assume that the graph of ${\displaystyle T}$ is closed and let ${\displaystyle {\mathcal {R}}:=T^{-1}:Y\rightrightarrows X.}$ It is easy to see that ${\displaystyle \operatorname {gr} {\mathcal {R}}}$ is closed and convex and that its image is ${\displaystyle X.}$ Given ${\displaystyle x\in X,}$ ${\displaystyle (Tx,x)}$ belongs to ${\displaystyle Y\times X}$ so that for every open neighborhood ${\displaystyle V}$ of ${\displaystyle Tx}$ in ${\displaystyle Y,}$ ${\displaystyle {\mathcal {R}}(V)=T^{-1}(V)}$ is a neighborhood of ${\displaystyle x}$ in ${\displaystyle X.}$ Thus ${\displaystyle T}$ is continuous at ${\displaystyle x.}$ Q.E.D.

Uniform boundedness principle — Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Fréchet spaces and ${\displaystyle T:X\to Y}$ be a bijective linear map. Then ${\displaystyle T}$ is continuous if and only if ${\displaystyle T^{-1}:Y\to X}$ is continuous. Furthermore, if ${\displaystyle T}$ is continuous then ${\displaystyle T}$ is an isomorphism of Fréchet spaces.

Proof

Apply the closed graph theorem to ${\displaystyle T}$ and ${\displaystyle T^{-1}.}$ Q.E.D.

Open mapping theorem — Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Fréchet spaces and ${\displaystyle T:X\to Y}$ be a continuous surjective linear map. Then T is an open map.

Proof

Clearly, ${\displaystyle T}$ is a closed and convex relation whose image is ${\displaystyle Y.}$ Let ${\displaystyle U}$ be a non-empty open subset of ${\displaystyle X,}$ let ${\displaystyle y}$ be in ${\displaystyle T(U),}$ and let ${\displaystyle x}$ in ${\displaystyle U}$ be such that ${\displaystyle y=Tx.}$ From the Ursescu theorem it follows that ${\displaystyle T(U)}$ is a neighborhood of ${\displaystyle y.}$ Q.E.D.

Corollary — Let ${\displaystyle X}$ be a barreled first countable space and let ${\displaystyle C}$ be a subset of ${\displaystyle X.}$ Then:

1. If ${\displaystyle C}$ is lower ideally convex then ${\displaystyle C^{i}=\operatorname {int} C.}$
2. If ${\displaystyle C}$ is ideally convex then ${\displaystyle C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.}$

Simons' theorem^[2] — Let ${\displaystyle X}$ and ${\displaystyle Y}$ be first countable with ${\displaystyle X}$ locally convex. Suppose that ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that ${\displaystyle X}$ is a Fréchet space and that ${\displaystyle {\mathcal {R}}}$ is lower ideally convex. Assume that ${\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}$ is barreled for some/every ${\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}$ Assume that ${\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}$ and let ${\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).}$ Then for every neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ in ${\displaystyle X,}$ ${\displaystyle y_{0}}$ belongs to the relative interior of ${\displaystyle {\mathcal {R}}(U)}$ in ${\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}$ (i.e. ${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}$). In particular, if ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }$ then ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}$

Robinson–Ursescu theorem^[3] — Let ${\displaystyle (X,\|\,\cdot \,\|)}$ and ${\displaystyle (Y,\|\,\cdot \,\|)}$ be normed spaces and ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ be a multimap with non-empty domain. Suppose that ${\displaystyle Y}$ is a barreled space, the graph of ${\displaystyle {\mathcal {R}}}$ verifies condition condition (Hwx), and that ${\displaystyle (x_{0},y_{0})\in \operatorname {gr} {\mathcal {R}}.}$ Let ${\displaystyle C_{X}}$ (resp. ${\displaystyle C_{Y}}$) denote the closed unit ball in ${\displaystyle X}$ (resp. ${\displaystyle Y}$) (so ${\displaystyle C_{X}=\{x\in X:\|x\|\leq 1\}}$). Then the following are equivalent:

1. ${\displaystyle y_{0}}$ belongs to the algebraic interior of ${\displaystyle \operatorname {Im} {\mathcal {R}}.}$
2. ${\displaystyle y_{0}\in \operatorname {int} {\mathcal {R}}\left(x_{0}+C_{X}\right).}$
3. There exists ${\displaystyle B>0}$ such that for all ${\displaystyle 0\leq r\leq 1,}$ ${\displaystyle y_{0}+BrC_{Y}\subseteq {\mathcal {R}}\left(x_{0}+rC_{X}\right).}$
4. There exist ${\displaystyle A>0}$ and ${\displaystyle B>0}$ such that for all ${\displaystyle x\in x_{0}+AC_{X}}$ and all ${\displaystyle y\in y_{0}+AC_{Y},}$ ${\displaystyle d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq B\cdot d(y,{\mathcal {R}}(x)).}$
5. There exists ${\displaystyle B>0}$ such that for all ${\displaystyle x\in X}$ and all ${\displaystyle y\in y_{0}+BC_{Y},}$ ${\displaystyle d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq {\frac {1+\left\|x-x_{0}\right\|}{B-\left\|y-y_{0}\right\|}}\cdot d(y,{\mathcal {R}}(x)).}$

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